At the time, Brown did not refer to any of the candidates by name, but his 50-minute speech was full of thinly veiled warnings that Corbyn would make Labour a party of protest rather than one of government.

He suggested that the MP for Islington North could damage international relations by allying with Hezbollah, Hamas, Venezuela and Russia.

It is not known why he has now decided to make his choice public after declining to reveal his favourite for the job a week ago. However, his preference for Cooper will not be seen as surprising given that he promoted her to the cabinet, and her husband, Ed Balls, was one of his closest allies.

In a speech on Tuesday, Cooper will attempt to woo young people enthused by Corbyn with a series of measures to tackle disillusionment in politics.

Cooper will unveil several new policies including a pledge to consult on moving the House of Lords out of London to a city such as Leeds, Manchester, Birmingham, Glasgow or Cardiff and a proposed freeze and cap on new appointments to the Lords until democratic reform is started.

She proposes getting rid of the “out of date and out of touch” way that MPs speak to each other in the House of Commons, which involves them referring to each other as “honourable” friends or members and showing their approval by crying “hear, hear”.

She would like to see electronic voting in parliament to end late-night sessions and to allow parents to take small children through the voting lobbies with them.

Meanwhile, Burnham gave one of his strongest warnings yet against his party taking a “wrong turn” and allowing itself to become disunited. Speaking in north London on Monday night, he argued that his party was at risk of losing the next two elections and being out of power for even longer than the 18 years under Margaret Thatcher and John Major.

“I am very clear that under my leadership, we will have the strongest opposition ever – taking on the Tories, getting after them day in, day out, setting out a clear and radical alternative. But we need more. Under my leadership, we will also build a party that will not just take the Tories on, but kick the Tories out,” he said.

“If we take the wrong turn now, we risk subjecting people to two decades of the Tories. We let that happen once before and I will fight with everything I’ve got to stop it happening again.”

Burnham said the party was in danger of “turning inward and talking to ourselves” during the leadership contest instead of addressing the whole nation. A source in his camp insisted that he was not talking about the danger of Corbyn in particular, but the risk of disunity under anyone but Burnham.

He claimed that for too long Labour had been afraid of using the word “tax”. “Let’s have a Labour alternative and a balanced plan for the economy that doesn’t reduce the deficit just by punishing the poorest and most valuable,” he said.

“A balanced plan where Labour isn’t afraid of using the word we never say in recent times. Tax! Why don’t we bring down the deficit by asking people who can to pay more tax.”

Burnham promised a church hall filled with supporters that his Labour party would not propose policies that were “hopeless dreams or pie in the sky”, but that they would “think big again”.

“Labour has spent too long bowing down before the mantra that the market has the answer to everything,” he said.

Introducing Burnham, the former deputy prime minister John Prescott told the audience that Labour had allowed itself to be “screwed” by Conservative propaganda on the economy.

“We didn’t put any argument up. We constantly argued to the PLP [parliamentary Labour party] and I was told by the two Eds that we were concentrating on the future and not the past,” said Prescott. “We got screwed on the past.”

He said of some of his former cabinet colleagues that they needed to “show a bit more humility when they make their comments about who they would work with or wouldn’t work with”.

Burnham was heckled by members of the audience demanding to know why he had voted with the party’s leadership and abstained on the welfare bill. Burnham responded that he had not wanted to lead a rebellion and split the party, adding that they were stronger united.

He addressed Corbyn’s pledge to apologise on behalf of Labour for the Iraq war, suggesting that he could be open to this after the Chilcot report was published if “apologies need to be made”.

Corbyn remains the firm favourite to win the contest, having galvanised the left with his anti-austerity politics and straight-talking manner.

On Monday, he set out his plans for improvement to mental healthcare, which included plans for schoolchildren to get education in emotional intelligence, life skills and parenting.

Earlier, he dismissed warnings that the party could split under his leadership, saying the number of MPs making “noises off” about him was “relatively small”.

Amid concerns among the candidates about the way the contest has been run, representatives of all campaigns will gather at Labour headquarters for a briefing on Tuesday.

In particular, Burnham’s camp made public their worries last week “about potential Tory infiltration on a large scale” among the party’s new 120,000 registered supporters, and raised the possibility of a legal challenge to the result if Corbyn won.

However, speaking to Channel 4 News on Monday, Burnham distanced himself from such a move, saying he would not personally seek to use the law to overturn the result, although others might.

He said he had confidence in the process as long as infiltrators were being identified, and claimed that a Tory councillor was at a Labour members’ leadership event, having already voted.

Sources in the campaigns said the briefing would be an opportunity for Labour to reassure the candidates that the contest was legally robust. They would want to know in particular that checks on new supporters were as thorough in areas without Labour representation as places where the local MP was helping to weed out names of known supporters of other parties.

They also want more details about the scale of the problem of infiltration, which Labour has been refusing to make public.

A party insider said there were now concerns at Labour’s central London headquarters that entryists might have wiped their Facebook and Twitter profiles of clues that they were supporters of other parties.

The source said it was not meant to be public knowledge that checks mostly involved monitoring the social media accounts of new members and supporters, but this information about the verification process had leaked out.

A Labour spokesman denied reports that 50,000 new members and supporters who signed up to vote in the contest could ultimately remain unverified before the winner was declared on 12 September.

It is understood that the party was so inundated with applications that it had to draft in a team of student volunteers to help with the process. However, the party is claiming that the bulk of the verification process will be completed by the end of this week.